Abstract
We present a general method to construct the sequence of new link polynomials and its two variable extension from exactly solvable models in statistical mechanics. First, we find representations of the braid group from the Boltzmann weights of the exactly solvable models. Second, we give the Markov traces associated with new braid group representations and using them construct new link polynomials. Third, we extend the theory into a two-variable version of the new link polynomials. Throughout the paper, we emphasize the essential roles played by the exactly solvable models and the underlying Yang-Baxter relation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Birman, J.S.: Braids, links, and mapping class groups. Princeton, NJ: Princeton University Press 1974
Rolfsen, D.: Knots and links. Berkeley, CA: Publish or Perish 1976
Artin, E.: Ann. Math.48, 101 (1947)
Alexander, J.W.: Proc. Natl. Acad. Sci. USA9, 93 (1923)
Markov, A.A.: Recueil Math. Moscov 73 (1935)
Jones, V.F.R.: Invent. Math.72, 1 (1983)
Jones, V.F.R.: Bull. Am. Math. Soc.12, 103 (1985)
Temperley, H.N.V., Lieb, E.H.: Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the “percolation” problem. Proc. Roy. Soc. Lond. A322, 251 (1971)
Lieb, E.H., Wu, F.Y.: In: Phase transitions and critical phenomena, Vol. 1, p. 331. Domb, C., Green, M.S. (eds.). London: Academic Press 1972
Baxter, R.J., Kelland, S.B., Wu, F.Y.: Equivalence of the Potts' model or Whitney polynomial with an ice-type model. J. Phys. A9, 397 (1976)
Bourbaki, N.: Groupes et algebres de Lie. Paris: Hermann 1968, Chap. 4
Alexander, J.W.: Trans. Am. Math. Soc.30, 275 (1928)
Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: Bull. Am. Math. Soc.12, 239 (1985)
Birman, J.S.: Invent. Math.81, 287 (1985)
Kanenobu, T.: Math. Ann.275, 555 (1986)
Faddeev, L.D.: Sov. Sci. Rev. Math. Phys. C1, 107 (1981)
Thacker, H.B.: Exact integrability in quantum field theory and statistical mechanics. Rev. Mod. Phys.53, 253 (1981)
Kulish, P.P., Sklyanin, E.K.: Lecture Notes in Physics, Vol. 151, p. 61. Berlin, Heidelberg, New York: Springer 1982
Wadati, M.: In: Dynamical problems in soliton systems, p. 68. Takeno, S. (ed.). Berlin, Heidelberg, New York: Springer 1985
Wadati, M., Akutsu, Y.: Exactly solvable models in statistical mechanics. In: Springer Series in Nonlinear Dynamics. Lakshmanan, M. (ed.). Berlin, Heidelberg, New York: Springer 1988
Karowski, M., Thun, H.J., Truong, T.T., Weisz, P.H.: On the uniqueness of a purely elasticS-matrix in (1+1) dimensions. Phys. Lett.67, 321 (1977)
Zamolodchikov, A.B., Zamolodchikov, A.B.: FactorizedS-Matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models. Ann. Phys. (NY)120, 253 (1979)
Sogo, K., Uchinami, M., Nakamura, A., Wadati, M.: Nonrelativistic theory of factorizedS-matrix. Prog. Theor. Phys.66, 1284 (1981)
Sogo, K., Uchinami, M., Akutsu, Y., Wadati, M.: Classification of exactly solvable two-component models. Prog. Theor. Phys.68, 508 (1981)
Baxter, R.J.: Exactly solved models in statistical mechanics. London: Academic Press 1982
Wu, F.Y.: Ising model with four-spin interactions. Phys. Rev. B4, 2312 (1971)
Kadanoff, L.P., Wegner, F.J.: Some critical properties of the eight vertex model. Phys. Rev. B4, 3989 (1981)
Zamolodchikov, A.B.:Z 4-symmetric factorizedS-matrix in two space-time dimensions. Commun. Math. Phys.69, 165 (1979)
Andrews, G.E., Baxter, R.J., Forrester, P.J.: Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities. J. Stat. Phys.35, 193 (1984)
Kuniba, A., Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn55, 1092, 2170, and 3338 (1986)
Kuniba, A., Akutsu, Y., Wadati, M.: An exactly solvable 4-state IRF model. Phys. Lett.116, 382 (1986) and An exactly solvable 5-state IRF model.117 A, 358 (1986)
Baxter, R.J., Andrews, G.E.: Lattice gas generalization of the hard hexagon model. I. Startriangle relation and local densities. J. Stat. Phys.44, 249 (1986)
Andrews, G.E., Baxter, R.J.: Lattice gas generalization of the hard hexagon model. II. The local densities as elliptic functions. J. Stat. Phys.44, 713 (1986)
Akutsu, Y., Kuniba, A., Wadati, M.: J. Phys. Soc. Jpn.55, 1466 and 1880 (1986)
Kuniba, A., Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.55, 2605 (1986)
Akutsu, Y., Kuniba, A., Wadati, M.: J. Phys. Soc. Jpn.55, 290 (1986)
Date, E., Jimbo, M., Miwa, T., Okado, M.: Fusion of the eight vertex SOS model. Lett. Math. Phys.12, 209 (1986)
Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.56, 839 (1987)
Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.56, 3039 (1987)
Sogo, K., Akutsu, Y., Abe, T.: New factorizedS-matrix and its application to exactly solvableq-state model. I and II. Theor. Phys.70, 730 and 739 (1983)
Zamolodchikov, A.B., Fateev, V.A.: A model of facterizedS-matrix and an integrable spin-1 Heisenberg chain. Sov. J. Nucl. Phys.32, 298 (1980)
Powers, R.T.: Ann. Math.86, 138 (1967)
Pimsner, M., Popa, S.: Preprint
Conway, J.H.: In: Computational problems in abstract algebra, p. 329. Leach, J. (ed.). London: Pergamon Press 1969
Akutsu, Y., Deguchi, T., Wadati, M.: J. Phys. Soc. Jpn.56, 3464 (1987)
Deguchi, T., Akutsu, Y., Wadati, M.: J. Phys. Soc. Jpn.57, No. 3 (1988)
Kauffman, L.H.: Preprint
Birman, J.S., Wenzl, H.: Preprint
Murakami, J.: Preprint
Murakami, J.: Preprint
Author information
Authors and Affiliations
Additional information
Communicated by A. Jaffe
Rights and permissions
About this article
Cite this article
Akutsu, Y., Wadati, M. Knots, links, braids and exactly solvable models in statistical mechanics. Commun.Math. Phys. 117, 243–259 (1988). https://doi.org/10.1007/BF01223592
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01223592